Electrochimica Acta 164 (2015) 97–107

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Electrochimica Acta

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A highly efficient reduced order electrochemical model for a largeformat LiMn2O4/Carbon polymer battery for real time applications

Yinyin Zhao, Song-Yul Choe *Mechanical Engineering, 1418 Wiggins Hall, Auburn Univ., AL 36849, USA

A R T I C L E I N F O

Article history:

Received 13 January 2015Received in revised form 19 February 2015Accepted 21 February 2015Available online 23 February 2015

Keywords:Lithium ion batteryelectrochemical modelingreduced order model

* Corresponding author. Tel.: +1 334 844E-mail addresses: yzz0031@tigermail.au

(S.-Y. Choe).

http://dx.doi.org/10.1016/j.electacta.2015.020013-4686/ã 2015 Elsevier Ltd. All rights

3329; faxburn.edu (

.182reserved.

A B S T R A C T

Mechanisms for ion transport, diffusion and intercalation/deintercalation processes in batteries duringcharging and discharging are described by governing equations that consist of partial differentialequations and nonlinear functions. Solving these equations numerically is computational intensive,particularlywhen the number of cells connected in series and parallel for high power or energy increases,whereas tolerance of errors should be kept under specified limits. Reduction of the computational time isrequired not only for enabling simulation of the behavior of packs, but also for development of a modelcapable of running in real time environments, so that new advanced estimation methods for state ofcharge (SOC) and state of health (SOH) can be developed.In our previous researchwork, a reduced ordermodel (ROM)was developed using different techniques

including polymoninal approximation and residue grouping, which represents the physical behaviors ofa battery. However, computational time has not been optimized. In this paper, methods to reduce thecomputational time are analyzed and employed to reduce the computational timewhile considering theaccuracy. Apart from retaining the residue grouping method for the ion concentration in electrolyte andlinearization of the Butler–Volmer equation, the Padé approximation is introduced to simplify thecalculation of ion concentration in electrode particles governed by the Fick’s second law. Meanwhile,discretized governing equations for potentials in electrodes and electrolytes are reduced by employingproper orthogonal decomposition (POD). In addition, expressions of the equilibrium potentials foranodes and cathodes are fitted to different order polynomials. The reduced equations are coupled toconstruct a single cell model for a large format lithium polymer battery that is validated againstexperimental data. The results show that the ROM proposed can reduce the computational time at leastto one-tenth of the models developed previously, while overall accuracies can be maintained.

ã 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Lithium ion batteries have been widely adopted as sources ofenergy storage for different power systems due to their high powerand energy density and reduced manufacturing costs triggered bythe growing market for electronic devices. However, improperoperating conditions such as continuous overcharge or under-charge can accelerate degradation processes and lead to earlyfailures. Model based battery management is an elegant approachthat enables monitoring of battery performances such as state ofcharge (SOC) or state of health (SOH), which are crucial for safeoperation of the battery.

: +1 334 844 3307.Y. Zhao), choe@auburn.edu

The models can be classified into two categories, equivalentcircuit models (ECM) and electrochemical thermal models (ETM)[1,2]. The ECM is easy to construct and fast in simulation, butcannot represent battery behavior completely [3]. Conversely, theETM based on electrochemical kinetics, mass balance, chargeconservation and thermal principles is capable of representing thebattery behavior accurately and providing internal variables likeion concentrations. However it is inappropriate for implementa-tion into microcontrollers that work in real time due to the largeamount of calculations caused by numerically solving discretizedpartial differential equations as well as nonlinear equations. TheETM describes the charge transport phenomenon in compositeelectrodes and electrolyte by a set of coupled partial differentialequations (PDEs). The PDEs are discretized and numerically solvedin radial and through-the-plane direction using the finite elementmethod (FEM) [4]. The resulting Full Order Model (FOM) provideshigh accuracy although the calculation of large amount statevariables is quite time consuming. In particular, the electrodes are

Nomenclature

A sandwich area of the cell (cm2)as specific surface area of electrode (cm�1)cs ion concentration in electrode phase (mol L�1)ce ion concentration in electrolyte (mol L�1)D diffusion coefficient (cm2 s�1)F Faraday constant (96,487Cmol�1)I current of the cell (A)i current density (A cm�2)i0 refference exchange current density (A cm�2)JLi reaction rate (A cm�3)L thickness of micro cell (cm)OCV open circuit voltage (V)R universal gas constant (8.3143 Jmol�1 K�1)Rs radius of spherical electrode particle (cm)r coordinate along the radius of electrode particle (cm)SOC state of chargeSOH state of healthSOX SOC, SOH, SOF(state of function), etc.T cell temperature (K)t time (s)Uequi equilibrium potential (V)x stoichiometric number in anodey stoichiometric number in cathodet0þ initial transference number

Greek symbolsa transfer coefficient for an electrode reactiond thickness (cm)e volume fraction of a porous medium’ Finite element method (FEM) solution of potentials’app approximate solution of potentials’e potential in electrolyte phase (V)’s potential in electrode phase (V)h surface overpotential of electrode reaction (V)k ionic conductivity of electrolyte (S cm�1) or kernel of

the data setkD concentration driven diffusion conductivity (A cm�1)s conductivity (S cm�1)

Subscriptsa anodicc cathodice electrolyte phaseequi equilibriummax maximum� negative electrode (anode)+ positive electrode (cathode)r radial direction in electrode particles solid phaseT terminal0% 0% SOC100% 100% SOC

Superscriptseff effectiveLi Lithium ion Table 1

Comparison of computational time (seconds) between FOM and previous ROM.

FOM Previous ROM [14]

Total 50.74 10.09Cs 19.25 6.05Ce 1.93 0.05Phi 7.01 3.36Others 22.55 0.63

98 Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107

modeled with spherical particles, and the evaluation of ionconcentrations in the particles should be carried out in the sphereradial direction that requires an extra discretization, whichmultiplies the number of state variables as well as thecomputational time.

Models for real-time control purpose should be able torepresent key physical phenomenawithin a reasonable time givenby the implementation. Therefore, the ETM should be simplifiedwhile considering the accuracy and computational time. Severalmodel order reduction techniques have been proposed in theliterature to reduce computational expense. Review of theliterature has shown that those methods can be sorted into twogroups: one is based on numerical calculations and the other one isbased on analytical expressions.

The numerical methods require discretization of the PDEs andreducing the matrix size by assorting eigenvalues of the system. Arepresentative method is the residue grouping (RG) method thatlumps states with similar eigenvalues, which reduces the order ofthe matrices [5]. Proper Orthogonal Decomposition (POD)technique approximates FOM behavior by a low order sub-modelderived from the most significant eigenvalues of matrices [6].

The analytical methods are replacements of the PDE for theFick’s second law that describes the ion diffusion process that takesplace in electrode particles. One of the analytical approachesemployed a parabolic profile based on volume average equation[7]. Accuracy is further improved using a quartic profile instead [8].Another analytical approach eliminated the independent spatialvariables in the diffusion equation by applying the pseudo steadystate (PSS) method, which is originated from finite integraltransform techniques [9]. In addition, the Galerkin method hasalso been adopted to reduce the computational complexity of theion concentration [10,11]. Furthermore, Padé approximation isapplied to the transfer function between current density and ionconcentrations [12], which is derived to study the diffusionimpedance in spheres [13].

The two groups of approaches aforementioned have their ownsuperiorities and limitations. The numerical methods are generallyapplicable for each part of the ETM and allow for systematicreduction of a model to a certain level that meets requirements foraccuracy. Numerical methods have better frequency responsescompared to the analytical expressions. However, the inevitablediscretization increases the complexity of the algorithms. Theanalytical expressions describe the model dynamics as a functionof time and space in an analytical closed form and avoiddiscretization of the PDEs. Nevertheless, those approximationsusually provide less accuracy in the high frequency range and thedetermination of the coefficients is quite time consuming,especially in the PSS method. In spite of the fact that the Padéapproximation is based on the analytical transfer function of theion diffusion impedance, it has excellent performance in a widefrequency range because the transfer functions are exact solutionsof the PDE. The features of high accuracy and efficiency enable it tobe a promising replacement of the diffusion equation in electrodeparticles.

In our previous research work for the development of ROM,order reductions and simplifications were performed, includingpolynomial approximation for ion concentration in electrodeparticles, residue grouping (RG) for ion concentrations in electro-lyte, and linearization of the Butler–Volmer equation [14]. Timeanalysis was carried out using the MATLAB profiler, as shown inTable 1, where the time measured in seconds indicates the

Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107 99

calculation duration for 1C discharge from 100% SOC to 0% SOCwith a sample rate of 1 s. The calculation time for the majorvariables in model are listed individually, where Cs, Ce and Phidenote the ion concentration in electrode particles, in electrolytes,and potential in electrodes and electrolytes, respectively. ROM canreduce computational time to approximately one-fifth of FOM,whereby Cs and Phi remain the most time consuming parts of thecalculation.

To reduce the calculation time for ion concentrations andpotentials, different techniques were investigated to simplify themodel. Four potential candidates for ion concentration in electrodeparticles governed by the Fick’s law are polynomial approximation(Poly), Padé approximation (Pade), POD and Galerkin Reformula-tion (GR), whose performances are compared and analyzed [15].Computational time and error of the four methods are plotted inFig. 1, where the time denotes the simulation duration of Cs,surfwhen a pulse current is applied for 1hour while the error is theaccumulated difference given by comparing them with a FEMsolution.

The results of the analysis showed that the performance of thePadé approximation was superior to the other options since thecalculation time remained within a reasonable tolerance and theerror quickly converged to zero when the order number increased.Moreover, the approximation allowed for a systematic reduction oforders dependent upon the level of accuracy required.

On the other hand, RG is retained for ion concentration inelectrolyte. The PODmethod as applied for reduction of large scaleODE systems is used to reduce the matrix size for potentials inelectrode and electrolyte.

2. Lithium ion battery model based on electrochemicalprinciples

A pouch type lithium ion polymer single cell is made of stackedmicrocells that are connected in parallel by current collectors. Dueto the high conductivity of the current collectors, the lateralcurrent flow from one microcell to another can be neglected andthe potentials on current collectors for eachmicrocell are assumedto be identical. Thus, the entire single cell can be regarded as amicrocell with only a pair of current collectors. The microcell has asandwich structure in the thickness direction that comprises ofcomposite electrodes mixed with electrolyte and a separator inbetween. The composite electrodes are made of active materials,

[(Fig._1)TD$FIG]

2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

Order

Tim

e/s

Poly

Pade

POD

GR

2 2.5 3 3.5 4 4.5 5

0

0.02

0.04

0.06

0.08

Order

Err

or

/ m

ol/

cm3 Poly

Pade

POD

GR

Fig. 1. Time and error analysis of four order reduction methods for ionconcentration in solid.

electrolytes and binders, where the particles dispersed onelectrodes are modeled as uniformly distributed spheres. Aschematic diagram for this cell is shown in Fig. 2, where theactive material on the anode is graphite and that on the cathode istransition metal oxides.

While cells discharge or charge, ions transport through theelectrolyte and react electrochemically at active materials, diffuse,and finally rest after intercalation in a lattice structure.Meanwhile,electrons flow through an external circuit and complete the redoxprocess. The working mechanism is described by a set of coupledgoverning equations. The equations incorporate ion and chargeconservation in electrode and electrolyte along with the Butler–Volmer equation that describes electrochemical kinetics. Fourvariables, including ion concentrations in the electrode and theelectrolyte, and potentials in the electrode and the electrolyte, thatappear in the governing equations are solved numerically using theFEM, which is called the Full Order Model (FOM). The governingequations are summarized in Table 2.

The OCV–SOC curve is obtained by pulse discharge, whereterminal voltage and the SOC are measured after the battery isdischarged with a small C rate for a short time and completelyrelaxed. The measured OCV is the difference between theequilibrium potentials for positive and negative electrode, wherethe potential for the negative electrode, U�, is approximated withan empirical equation, as provided in Table 2. Then, the potentialfor the positive electrode, U+, is fitted by a high order polynomial.The equilibrium potentials versus stoichiometric numbers areplotted in Fig. 3.

3. Model order reduction and simplification

Coupled governing equations consist of nonlinear equationsand PDEs that are numerically solved by the FEM in FOM. Thereductions and simplifications of the equations for each part of themodel are described below.

3.1. Ion concentration in electrodes

Ion diffusion in a particle is described by the second order Fick’slaw, whose solution provides gradients of ion concentration, Cs,along the radial direction in electrode particles. For controloriented models, the ion concentration on particle surface, Cs,surf,and the volume average concentration, Cs,ave, are critical variablesbecause of their direct relationship to the reactions that affectintercalation in active materials and SOC. Analytical solutions ofthe Fick’s law provide two transfer functions where the currentdensity, JLi, and two ion concentrations, Cs,surf,Cs,ave, are regarded asan input variable and output variables, respectively, as shown asfollows [12]:

Cs;surf ðsÞjLiðsÞ

¼ 1asF

Rs

Ds

� �tanhðbÞ

tanhðbÞ � b

� �(1)

where b ¼ Rsffiffiffiffiffiffiffiffiffiffis=Ds

p.

Cs;aveðsÞjLiðsÞ

¼ � 3RsasFs

(2)

The Qth order Padé approximation for Cs,surf in Eq. (1) results in theform of:

Cs;surf ðsÞjLiðsÞ

¼ a0 þ a1sþ a2s2 þ � � � aQ�1sQ�1

sð1þ b2sþ b3s2 þ � � � bQsQ�1Þ (3)

The coefficients a0; a1; � � � ; aQ�1 and b2; b3; � � � ; bQ are determinedby comparing the derivatives of Eq. (1) with Eq. (3) at s = 0. Theresulting low order Padé approximations are listed in Table 3,where m ¼ 1

asFDs.

[(Fig._2)TD$FIG]

Fig. 2. Schematic diagram of a model for a pouch type single cell.

100 Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107

A state space representation in controllable canonical form ofthe transfer function in Eqs. (2) and (3) is shown as:

_~x ¼ A1~x þ B1jLi ~y ¼ C1~x (4)

where

A1 ¼

0 1 0 0 � � � 00 0 1 0 � � � 00 0 0 1 � � � 0... ..

. ... ..

.} 0

0 0 0 0 � � � 1

0 � 1bQ

�b2bQ

�b3bQ

� � � �bQ�1

bQ

2666666664

3777777775; B1 ¼

000...

01

26666664

37777775;

x ¼

x1x2x3...

xQ�1xQ

266666664

377777775C1 ¼

a0bQ

a1bQ

a2bQ

a3bQ

� � � aQ�1

bQa0bQ

a0b2bQ

a0b3bQ

a0b4bQ

� � � a0

2664

3775;

~y ¼ Cs;surfCs;ave

� �(5)

Initial conditions for solving the equations above are steady state,_x ¼ 0, zero input jLi ¼ 0 and uniform concentration, Cs;surf ¼ Cs;ave.The first two conditions result in a linear equation, A~x ¼ 0, thatyields a solution with one degree of freedom,~x0 ¼ x0 0 0 � � � 0 0½ �T . The third condition gives an initialvalue of the concentration, Cs;surf ¼ Cs;ave ¼ Cs;0. As a result, theinitial state is presented as follows:

Table 2Governing equations for ETM.

Conservation equations

Ion conservation in electrode @Cs@t ¼ Ds

r2@@r r2@Cs

@r

� �Ion conservation in electrolyte @ðeeCeÞ

@t ¼ @@x Def f

e@@xCe

� �þ 1�t0þ

F jLi

Charge conservation in electrode @@x s

ef f @@xfs

� jLi ¼ 0

Charge conservation in electrolyte @@x k

ef f @@xfe

þ @@x kef f

D@@xlnCe

� �þ jLi ¼ 0

Electrochemical kinetics

Butler–Volmer equation jLi ¼ asi0 exp aaFRT h

� �� exp �acFRT h

� � �

Equilibrium potentials

U�ðxÞ ¼ �1:8636x5 þ 6:6478x4 � 9:07126x3 þ 5:7843x2 � 1:79UþðyÞ ¼ �1550:039y7 þ 6806:298y6 � 12633:679y5 þ 12829:3

2715:077y2 � 523:437yþ 46:682x ¼ Cs;surf� =Cs;max�y ¼ Cs;surfþ =Cs;max�

~x0 ¼ Cs;0bQa0

0 0 � � � 0 0� �T

(6)

3.2. Ion concentration in electrolyte

The RG method is used to reduce the PDE that describes iontransport in electrolytes, where current density across eachelectrode is assumed to be uniform, as shown in Eq. (7):

jLi� ¼ IAL�

ðNegative electrodeÞjLisep ¼ 0 ðseparatorÞjLiþ ¼ � I

ALþðPositive electrodeÞ

(7)

The PDE is discretized by the FEM and the resulting equations areconverted into the state space representation:

_~x ¼ A2~x þ B2I y ¼ C2~x þ D2I (8)

The equation above is a single input M output linear system withmatrices A2;B2;C2andD2 that have dimensions ofM �M;M � 1;M �M;M � 1, where M denotes the number ofdiscrete node points along the microcell thickness direction. Theinput variable is the applied current I and the output is aM�1 vector that indicates ion concentration, Ce, at each nodepoint.

70xþ 0:329427y4 � 7687:388y3þ

Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107 101

The state equation is transformed to a modal canonical formwith:

A2 ¼ diag½l1 l2 � � � lM �B2 ¼ ½1 1 � � � 1 �T

C2 ¼ ½ r1 l1 r2 l2 � � � rM lM�

D2 ¼ Z þXnk¼1

rk!

" #(9)

wherelk is the eigenvalues of the systemmatrix, A2, Z is the steady

state vector evaluated by Z ¼ �C2A�12 B2 þ D2 and~rk is the residue

vector obtained by:

~rk ¼C2~qk~pkB2

lk(10)

~qk and ~pk denote the right and left eigenvectors of the systemmatrix A2 and A2~qk ¼ lk~qk, ~pkA2 ¼ lk~pk.

The Mth order system is reduced to the Nth order by groupingthe residues according to the similarity of the eigenvalues [5]. Thegrouped residue vectors and the grouped eigenvalues are definedas:

~rf ¼Xkf

k¼kf�1þ1

~rk

lf ¼

Xkfk¼kf�1þ1

lk~rk

~rf(11)

The Nth order state equation is constructed with the reducedmatrices A�

2;B�2;C

�2;D

�2 that have a dimension of

N � N;N � 1;M � N;M � 1:

A�2 ¼ diag½l1 l2 � � � lN �

B�2 ¼ ½1 1 � � � 1 �T

C�2 ¼ ½ r1 l1 r2 l2 � � � rN lN�

[(Fig._3)TD$FIG]

0.30.40.50.60.70.80.91

3

3.5

4

4.5

y = Cs,surf + / Cs,max +

U+ /

V

Equilibrium potential in composite cathode

OCV

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

x = Cs,surf - / Cs,max -

U- /

V

Equilibrium potential in composite anode

Fig. 3. The OCV curve obtained experimentally and the equilibrium potentials ofelectrodes.

D�2 ¼ Z þ

XNf¼1

rf!

24

35 (12)

3.3. Potentials in electrodes and electrolytes

POD is a mathematical procedure that is used to find a basis fora modal decomposition of a data set. The data applied here is therigorous solution of the potentials in electrode and electrolyteobtained by FEM. The governing equations for charge conservationin electrode and electrolyte are discretized and the resultingequations are expressed in the matrix form as follows:

A3~f ¼ b (13)

where ~f is the FEM solution vector of the potentials at each gridpoints, A3 and b denote the coefficients matrix and constant vector.

~f can be approximated by a linear combination of the first N

proper orthogonal modes (POMs) as shown in Eq. (14), where~fappr

is the approximation solution,F is the POMs and a is the reductionvariables.

~fappr ¼ F �~a (14)

The POMs are the eigenvectors obtained by singular value

decomposition of the discrete kernel given by ~k ¼ ~fT � ~f and

sorted in a manner that the corresponding eigenvalues are in anon-increasing order, as shown in Eq. (15).S is the diagonalmatrixof eigenvalues and N denotes the model order that we applied inPOD.

USV� � ¼ svdð~f!

fullT � ~f

!full Þ F ¼ Uð:;1 : NÞ

The reduction variables,~a, are solved by substituting Eq. (14) intoEq. (13), which yields Eq. (16):

A3 F �~a ¼ b~a

¼ FTA�13 b (16)

Table 3Low orders Padé approximation of Cs,surf.

Order Padé approximation

2nd

�3DsmRs

� 2mRss7

sþ R2s s2

35Ds

3rd

�3DsmRs

� 4mRss11 � mR3

s s2

165Ds

sþ 3R2s s2

55Dsþ R4

s s3

3465D2s

4th

�3DsmRs

� 2mRss5 � 2mR3

s s2

195Ds� 4mR5

s s3

75075D2s

sþ R2s s2

15Dsþ 2R4

s s3

2275D2sþ R6

s s4

675675D3s

5th

�3DsmRs

� 8mRss19 � 21mR3

s s2

1615Ds� 4mR5

s s3

33915D2s� mR7

s s4

3968055D3s

sþ 7R2s s2

95Dsþ 3R4

s s3

2261D2sþ 2R6

s s4

305235D3sþ R8

s s5

218263025D3s

102 Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107

The selected POMs form a basis that facilitates to capture thedominant characteristics of the system dynamics and enablesrepresentation of the most significant information of the variables

under various circumstances. In this work, the potential data set ~fused to build the kernel ~k is evaluated under 1C dischargecondition with uniform current density.

3.4. Electrochemical kinetics

Electrochemical reactions that take place at the interfacebetween electrodes and electrolytes are described by theButler–Volmer equation. The nonlinear functions are linearizedbased on the fact that the overpotentials vary within a linear rangeunder normal operating conditions [4].

jLi ¼ asi0ðaa þ acÞF

RTh (17)

4. Analysis of sub-models performances

Performances of the sub-models are analyzed by comparing themodels before and after reduction. Themodel before the reductionis numerically solved using the finite element method (FEM),where the number of grids in the radial direction of particles andthickness direction of the cell are 25 and 15, respectively. The threevariables, Cs;surf , Ce and f are used to evaluate performances of thesub-models.

4.1. Order reduction for ion concentration in electrode particles

Since the average concentration in electrode particles, Cs;ave, iseasily obtained by the volume average equation, only the surfaceconcentration, Cs;surf , is the crucial variable to be considered for theevaluation of performances with respect to frequency, time and

static response. The response of Cs;surf at a current density, jLi, canbe formulated by different methods including parabolic, quarticpolynomial and the Padé approximation. The transfer functionsreformulated from a parabolic and a quartic profile are shown inEq. (18) [10].

[(Fig._4)TD$FIG]

10-3

10-2

10-1

100

101

102

103

-150

-100

-50

Exact

2nd

10th

Parabolic

Quartic

ω

ω

(Rad/sec)

Mag

nit

ud

e (d

b)

10-3

10-2

10-1

100

101

102

103

80

100

120

140

160

180

Exact

2nd10th

Parabolic Quartic

(Rad/sec)

Ph

ase

Ab

gle

(º)

Fig. 4. Frequency responses of the surface concentration of a particle on the anodeside: analytical exact solution, parabolic and quartic polynomial and the differentorder of Padé approximation.

Cs;surf ðsÞjLiðsÞ

¼ �15mDs þ R2s ms

5Rssparabolic

Cs;surf ðsÞjLiðsÞ

¼ �3150D2s mþ 315DsmR2

s sþmR4s s

2

35RssðR2s sþ 30DsÞ

quartic

wherem ¼ 1asFDs

(18)

Frequency responses of the analytical exact solution given inEq. (1), the two different polynomial approaches and the Padéapproximation from the 2nd to 10th order with a step of a two-order for surface concentration of particles on the anode side areplotted in Fig. 4. Comparison between the analytical solution andthe two approximation methods shows that the Padé approxima-tion approaches the analytical exact solution up to a certainfrequency that depending upon the order. The higher the order is,the smaller the difference of the responses in magnitude andphase. In addition, the range of the response of the 4th order Padéapproximation is comparable to that of the quartic polynomial.Conversely, the parabolic polynomial only works over the range ofvery low frequencies, which means that the dynamic performanceis of less satisfactory. The quartic profile with an extra state betterrepresents its response over a high frequency range, but it is hard toextend the order of polynomial systematically tomeet the accuracyrequirements.Time responses of Cs;surf by applying differentmethods at AC currents are compared in Fig. 5. Relative errorsare calculated in percentage, which are shown in Eq. (19), where yiis the simulation data of the different methods, and yi is the FEMsolution with 25 grid points on radial direction (or experimentaldata). The parabolic method is not considered because of its largeerror.

Relative error ¼ yi � yiyi

� �� 100% (19)

The 2nd order of the Padé approximation (yellow dot-dashed line)shows the largest discrepancy in magnitude and phase comparedto the FEM solution. The quartic profile (purple dashed line) showsimproved tracking behavior, while the 6th order of the Padéapproximation (green line) has the closest match to the FEMsolution. The time responses shown in Fig. 5 verify the results ofthe frequency responses as shown in Fig. 4 where the Padéapproximation can best represent the FEM solution over a widefrequency range if an appropriate order is determined.

[(Fig._5)TD$FIG]

0 20 40 60 80 100 120 140 160 180 200

0.9

0.95

1

1.05

1.1

time/s

Cs,

surf

/ C

s,0

FEM

Quartic

2nd

6th

0 20 40 60 80 100 120 140 160 180 200-10

-5

0

5

10

Quartic2nd

6th

time/s

Rel

ativ

e er

ror

(%)

Fig. 5. Time responses of surface concentration of a particle on anode: FEM, quarticpolynomial, and 2nd and 6th order of the Padé approximation.

[(Fig._6)TD$FIG]

0 1000 2000 3000 4000 5000 6000 70000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time/s

x =

Cs,

surf

- / C

s,m

ax-

0.5C FEM

0.5C 3rd order Pade

1C FEM

1C 3rd order Pade

2C FEM

2C 3rd order Pade

3C FEM

3C 3rd order Pade

Fig. 6. Stoichiometry number of the anode during full discharge: 0.5 C, 1 C, 2 C and3C rate.

[(Fig._8)TD$FIG]

0 10 20 30 40 50 6010

-15

10-10

10-5

100

105

Index of Eigenvalues, i

Eig

env

alu

es, λ

Fig. 8. Eigenvalue spectrum of the potential data set obtained from the model withFEM during discharge at 1C rate.

Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107 103

One other criterion for assessments is the accuracy of thesurface concentration at steady state. According to the analysis inFig. 4, the 3rd order Padé approximation should be enough torepresent steady state or low frequency behavior of variables.Simulation results of the stoichiometry number in the anode sideduring the discharging process at different C rates are depicted inFig. 6, where the stoichiometry number is a function of surfaceconcentration that is defined as x ¼ Cs;surf�=Cs;max� and is used todetermine the equilibrium potential in the anode. The resultsobtained by the 3rd order Padé approximation are compared tothose given by FEM (circles), which shows that the 3rd order Padéapproximation is accurate enough to replace the FEM method inthe steady state. The model order should be determined bybalancing the accuracy and the computational time [14].

4.2. Order reduction for ion concentration in electrolyte

The 3rd order of residue grouping (RG) method is employed toreplace the FEMmethod for calculation of ion concentrations in theelectrolyte. Distributions of ion concentrations using RG and FEMat various time instants are plotted in Fig. 7, where the battery isdischarged from 100% SOC with 1C rate. The x and y coordinatesrepresent dimensionless thickness and ion concentration,

[(Fig._7)TD$FIG]

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

0s 1s 2s

5s

10s

20s200s

x/L

Ce/C

e,0

FEM

3rd order RG

Fig. 7. Ion concentration in the electrolyte at various times during 1C discharge.

respectively. The nominated Ce=Ce;0 with circles are the concen-trations calculated by FEM at each grid point across the cell andthose with solid lines are calculated by the 3rd order RG sub-model.With uniform current density applied, the results show thatthe RG method can follow the response of FEM with a decentaccuracy, which can be further improved by appropriatelyadjusting the upper and lower limits of each eigenvalue group.

4.3. Order reduction for potentials in electrode and electrolyte

POD is employed to reduce the order for potentials. Todetermine the order for POD, eigenvalue spectrum of the discrete

kernel, ~k ¼ ~ffullT � ~ff ull, is calculated and plotted in Fig. 8, where

the eigenvalues, li are the diagonal elements of matrix S inEq. (15). The results show that the magnitude rapidly decreases forthe first four eigenvalues and then remains a small constant value.Since the importance of the POMs is denoted by the correspondingeigenvalues, the first 4 POMs can represent the most significantdynamic characteristics of the system. As a result, the basis of thePOD can be determined by considering the eigenvalue spectrum,and a linear combination of the selected POMs is used to constructa reduced model for potentials in electrode and electrolyte. In thiscase, a 3rd order sub-model is employed.

[(Fig._9)TD$FIG]

00.10.20.30.40.50.60.70.80.91-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

SOC

Po

ten

tial

/ V

0.5C FEM

0.5C 3rd order POD

1C FEM

1C 3rd order POD

2C FEM

2C 3rd order POD

3C FEM

3C 3rd order POD

Fig. 9. Electrolyte potential in the center of the cell during dischargingwith 1C rate.

[(Fig._10)TD$FIG]

00.10.20.30.40.50.60.70.80.912.8

3

3.2

3.4

3.6

3.8

4

4.2

SOC

Po

ten

tial

/ V

0.5C FEM

0.5C 3rd order POD

1C FEM

1C 3rd order POD

2C FEM

2C 3rd order POD

3C FEM

3C 3rd order POD

Fig. 10. Electrode potential at the interface between separator and compositecathode during discharge with 1C rate.

104 Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107

To assess performance of the 3rd order POD sub-model,potentials in electrode and electrolyte calculated by reduced ordersub-models are compared to those of FEM in Figs. 9 and 10, wherethe battery is discharged with 0.5 C, 1 C, 2 C and 3C rate from 100%down to 0% SOC.

Since the data produced by the reduced sub-model and FEMresults are hard to compare using a 3D surface plot, only twovariables at representative grid points are selected to access theaccuracy of the calculations. Variation of the electrolyte potentialin themiddle of the cell is shown in Fig. 9, where the negative valueof potential is caused by the zero potential refference that is set atthe anode boundary near the current collector. The electrodepotential at the cathode-separator interface point is also given inFig. 10.

Comparison shows that the responses of the 3rd order of PODsub-model (solid curves) are comparable to those of the FEM(circles) that has 15 grid points along the thickness directionincluding three grid points on the separator. The proposingmethodallows for substantial reduction of the matrix size of the potentialfrom 27�27 to 3�3, so that the computational time can besignificantly reduced while maintaining the accuracy.

5. ROM for a single cell

ROM for a single cell is constructed by the sub-models. Aschematic diagram for the integrated ROM is shown in Fig. 11,where input variables are load profiles along with ambient

[(Fig._11)TD$FIG]

Fig. 11. Schematic diagram for

temperature that can be constant current (CC) or constant voltage(CV) charging and discharging. The data set for potential ’ iscalculated offline using FEM and then stored for POD. Parametersused for these specific cells are listed in the appendix. Order forsub-models is defined as an extra input parameter that can bedetermined upon required accuracy. Systemmatrices for each sub-model is determined based on parameters, initial and boundaryconditions, and stored prior to the main loop.

The main loop consists of three sub-models. Ion concentrationsare firstly solved with uniform current density, the resulting statevariables Cs;surf , Cs;ave and Ce are used to determine the equilibriumpotential, SOC and diffusion term in charge conservation equationscorrespondingly. Then the potentials are calculated based on theion concentrations at the instants.

The output variables are terminal voltage or current, ionconcentrations, SOC and other internal variables.

5.1. Effects of sampling time

Sampling time is one of the crucial factors for the determinationof hardware. A short sampling time can overload hardware andincrease its overall costs, particularly when a large number of cellsare connected in series and parallel. Therefore, computational timeand error of terminal voltage is calculated as a function of samplingtimes and plotted in Fig.12, where the initial SOC of the batterywas100% and the battery was discharged with 1C rate. The model iscoded with MATLAB and runs on PC with 3.4GHz processor,whereas execution time is measured. The errors are normalized asbelow, where yi and yi are the data obtained from simulations andexperiments and n is the number of sampling points.

Normalized error ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP ðyi � yiÞ2

n

s(20)

The blue line with stars and green line with points denote thecomputational time and errors, respectively. As expected, theerrors are reciprocal proportional to the computational time. Anoptimal sampling time can be determined by considering time andaccuracy. We chose about 1 second as the optimal sampling time,where the curves cross.

5.2. Experimental validation of the ROM

The constructed ROM is validated against experimental datacollected from a test station that was constructed with aprogrammable power supply and an electronics load that arecontrolled by LabVIEW. The cells used for this validation are pouchtype lithium polymer batteries, whose specifications are listedbelow.

the ROM for a single cell.

[(Fig._14)TD$FIG]

0 1000 2000 3000 4000 5000 6000 7000 8000

3

3.2

3.4

3.6

3.8

4

4.2

time/s

Ter

min

al v

olt

age

/ V

0.5C Exp

0.5C ROM

1C Exp

1C ROM

2C Exp

2C ROM

3C Exp

3C ROM

Fig. 14. Comparison of terminal voltage from simulation and experiments during0.5 C, 1 C, 2 C and 3C charge.

[(Fig._12)TD$FIG]

10-1

100

101

102

0

5

10

Co

mp

uta

tio

nal

tim

e /

s

sample time / s10

-110

010

110

20

2

4x 10

-3

No

rmal

ized

err

or

/ V

time

error

Fig. 12. Computational time and error with respect to different sample rate.

Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107 105

Chemistry: Cathode, LiMn2O4 (spinel); Anode, surface modifiedgraphite; Electrolyte, Gel polymer (LiPF6 + EC/DEC/EMC);Separator, Ceramic coated separator.

Dimension: 14.7mm�280mm�185mm.Nominal capacity: 50Ah.Operation range of the terminal voltage: 2.7V–4.2V.The experimental data are acquired under constant tempera-

ture at 25 �C by using an extra thermal chamber and with differentcurrent loads that include full charge and full discharge withconstant current at 0.5, 1, 2 C, 3 C rate, continuously charging anddischarging multiple cycles and a current profile obtained from adriving cycles of an electric vehicle. Since the nominal capacity ofthe large format lithium ion battery tested in this work is 50Ah,which is relatively high compared to other pouch type batteries,the ROM was validated up to 3C. For continuous charging anddischarging at higher C-rate, severe heat generation caused bylarge current will accelerate the battery degradation and result inchanges of physical parameters in the ROM, which obstacle thevalidation of the model.

5.2.1. Discharging and chargingTerminal voltage Vt between simulation results of ROM (solid

line) and experimental data (circles) are plotted in Figs. 13 and 14,

[(Fig._13)TD$FIG]

0 1000 2000 3000 4000 5000 6000 7000

2.8

3

3.2

3.4

3.6

3.8

4

4.2

time/s

Ter

min

al v

olt

age

/ V

0.5C Exp

0.5C ROM

1C Exp

1C ROM

2C Exp

2C ROM

3C Exp

3C ROM

Fig. 13. Comparison of terminal voltage from simulation and experiments during0.5 C, 1 C, 2 C and 3C discharge.

where the simulation results of ROM have a fairly goodmatch withthe experimental data at different current rates.

The percentage of relative errors defined in Eq. (19) duringdischarging and charging are plotted in Fig. 15. Overall errors areless than 2% except at the end of charging and the beginning ofdischarging where the SOC is relatively low. The error is caused bythe simplified equation for the overpotential since the terminalvoltage is the difference between the OCV and the overpotentialthat becomes large at low SOC range.

5.2.2. Multiple cycles and EV driving cycleLong term stability of the ROM is assessed by the response of

terminal voltage at multiple cycles and an EV driving cycle. Themultiple cycles consist of a combination of CC/CV charging and CCdischarging with various current amplitudes. Simulation results ofthe ROM are compared to the experimental data in Figs. 16 and 17.There are some transient errors appeared at the instant when anabrupt current change occurs, which is caused by inaccuratedetermination of equilibrium potential that is a function of surfaceconcentration as shown in Table 2. In fact, the surface concentra-tion in electrode particle is estimated by 3rd order Padéapproximation, which is accurate enough to represent the staticresponses as shown in Fig. 6, but is inadequate to capture the high

[(Fig._15)TD$FIG]

0 1000 2000 3000 4000 5000 6000 7000

-2

0

2

time/s

Rel

ativ

e er

ror

/ (%

) discharge error

0.5C

1C

2C

3C

0 1000 2000 3000 4000 5000 6000 7000

-2

0

2

time/s

Rel

ativ

e er

ror

/ (%

) charge error

0.5C

1C

2C

3C

Fig. 15. Percentage errors of the terminal voltage at discharge and charge.

[(Fig._18)TD$FIG]

0.1 1 10 1200

20

40

60

80

100

sample time / s

Co

mp

uta

tio

nal

tim

e /

s

previous ROM

new ROM

0.1 1 10 1200

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3

sample time / s

No

rmal

ized

err

or

/ V

previous ROM

new ROM

Fig. 18. Comparison of computational time and error vs. sample time between theprevious ROM [14] and the new proposing ROM.

Table 4Comparison of computational time (second) between previous ROM and the newapproach.

Previous ROM [14] New ROM

Total 10.09 1.05Cs 6.05 0.08Ce 0.05 0.03Phi 3.36 0.34Others 0.63 0.6

[(Fig._16)TD$FIG]

0 2 4 6 8 10 12

-100

-50

0

50

100

time/h

Cu

rren

t /

Am

p

0 2 4 6 8 10 122.5

3

3.5

4

4.5

time/h

Ter

min

al V

olt

age

/ V

ROM

Exp

Fig. 16. Comparison of terminal voltage between simulation and experiments atmultiple charging and discharging cycles.

106 Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107

frequency dynamics presented in driving cycles. Since the order ofthe Padé approximation is systematically adjustable unlike thepolynomial approach, the accuracy of ROM can be furtherimproved by increasing the order of the approximation, whichleads to high computational expense.

5.2.3. Comparison with the ROM developed previouslyTwomajor features of the newly developed ROM are compared

with those of the previously developed ROM, which includecomputational time and normalized voltage errors as a function ofthe sampling time, as plotted in Fig. 18. The computational time issignificantly reduced for the sampling time that is less than 1 secwhile the errors of terminal voltage are almost the same until70 sec, but are less after 70 sec.

Finally, execution time of three models including the timerequired for calculation of sub-models is listed in Table 4. The timeanalysis was carried out using the MATLAB profiler and theindividual termwas calculated based on the execution time of eachsub-model functions. The ROM proposed in this paper needs theshortest time and reduces the computation time to one tenth of theprevious ROM.

[(Fig._17)TD$FIG]

0 100 200 300 400 500 600 700 800

0

20

40

60

time/s

Cu

rren

t /

Am

p

0 100 200 300 400 500 600 700 8004

4.05

4.1

4.15

4.2

time/h

Ter

min

al v

olt

age

/ V

Exp

ROM

Fig. 17. Comparison of terminal voltage from simulation and experiments at acurrent profile measured at an EV driving cycle.

6. Conclusion

Electric equivalent circuit models have been widely used foralgorithms of SOXs embedded in BMS, but have limitationsbecause of the absence of the physical phenomena that take placein batteries. ROM based on electrochemical principles can providea potential solution that allows for monitoring internal physicalvariables. As a result, advanced control algorithms can be designedbased on these variables. In fact, hundreds of cells are needed forhigh power systems, where the computational time of themodel isone of impeding factors for implementations on microcontrollers.

This paper proposed a ROM that is constructed by integration ofsub-models derived using different order reduction techniques.The diffusion in solid is simplified by Padé approximation, whilethe concentration in electrolyte is reduced by applying the residuegrouping method. In addition, POD is adopted to shrink the matrixsize for potentials. The integrated ROM is then validated againstexperimental data at various operating conditions. Main accom-plishments and findings are summarized as follows:

�

Comparative analysis of the performances of the individualreduced sub-models in time and frequency domain,

�

Selection of sampling time based on errors, � Performance of the developed ROM� Reduction of computational time to one-tenth of the previousROM while the accuracy remains the same or better than theprevious ROM regardless of the sampling time.

� The order is adjustable dependent upon input profiles or

accuracy requirements.

Y. Zhao, S.-Y. Choe / Electrochimica Acta 164 (2015) 97–107 107

Future work will include development and integration of thedegradation model. In addition, the accuracy of the model in realtime operation will be improved using advanced control methods.

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